Subgroups — #10711560

Order 768.

302 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#1848	C₂	1	1	yes	maximal, characteristic
2	#69764	C₂	1	1	yes	maximal, characteristic
3	C₄×#812	C₂	1	1	yes	maximal, characteristic
4	#67730	C₂	1	4	yes	maximal
5	#12886	—	3	1	no	maximal, Sylow
6	C₂×C₃×#108	—	2	1	no	—
7	C₂×C₃×#108	C₂²	1	2	yes	—
8	C₃×#423	—	2	1	no	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×#817	—	2	1	no	—
11	C₂×#817	C₂²	1	2	yes	—
12	C₂×#812	C₂²	1	2	yes	—
13	#4935	—	2	1	no	—
14	#4461	C₄	1	2	yes	complemented
15	C₄×C₃⋊C₁₆	C₄	1	2	yes	complemented
16	C₄×M₃₂	—	3	1	no	—
17	#1848	D₃	1	1	yes	characteristic
18	#1844	—	3	1	no	—
19	#1383	—	3	4	no	—
20	C₂×C₃×M₁₆	—	2	2	no	—
21	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
23	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
24	C₂×C₄⋊Dic₃	—	2	2	no	—
25	C₂×C₄⋊Dic₃	—	4	2	no	—
26	C₃×#108	—	2	4	no	—
27	C₃×#108	—	4	2	no	—
28	C₃×C₄⋊C₈	—	4	2	no	—
29	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
32	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
33	#817	—	2	4	no	—
34	#817	—	4	2	no	—
35	#812	—	2	2	no	—
36	#812	—	2	2	no	—
37	C₂×C₃⋊C₁₆	C₂×C₄	1	4	yes	—
38	C₂×M₃₂	—	3	2	no	—
39	C₂×#108	—	2	1	no	—
40	C₂×#108	C₂×D₃	1	2	yes	—
41	C₂×#106	—	3	2	no	—
42	C₂×#106	—	6	1	no	—
43	#423	—	2	1	no	—
44	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
45	#384	—	6	1	no	—
46	#346	—	3	2	no	—
47	C₄×C₁₆	—	3	2	no	—
48	C₂²×C₃×C₄	SD₁₆	1	2	yes	—
49	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₃	—	4	2	no	—
51	C₃×M₁₆	—	4	2	no	—
52	C₃×M₁₆	—	4	2	no	—
53	C₂×C₃×C₈	—	4	2	no	—
54	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₃×C₈	D₈	1	2	yes	—
57	C₂×C₃×C₈	Q₁₆	1	2	yes	—
58	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₃×C₈	#41	1	2	yes	—
61	C₂×C₃×C₈	#41	1	2	yes	—
62	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
63	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
64	C₃×C₄²	SD₁₆	1	2	yes	—
65	C₄⋊Dic₃	—	4	2	no	—
66	C₄⋊Dic₃	—	4	2	no	—
67	C₄⋊Dic₃	—	8	4	no	—
68	C₃⋊C₁₆	—	2	4	no	—
69	C₂×M₁₆	—	2	2	no	—
70	C₂²×C₈	#67	1	1	yes	characteristic
71	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
72	C₂×C₄⋊C₄	—	6	2	no	—
73	C₂×C₄⋊C₄	—	12	2	no	—
74	C₂×C₄²	#67	1	1	yes	characteristic
75	M₃₂	—	6	2	no	—
76	M₃₂	—	6	2	no	—
77	C₂×C₁₆	—	3	4	no	—
78	#108	—	2	4	no	—
79	#108	—	4	2	no	—
80	#106	—	6	4	no	—
81	#106	—	12	2	no	—
82	C₄⋊C₈	—	4	2	no	—
83	C₄×C₈	Dic₆	1	1	yes	characteristic
84	C₄×C₈	Dic₆	1	1	yes	characteristic
85	C₄×C₈	C₄×D₃	1	1	yes	characteristic
86	C₄×C₈	C₄×D₃	1	1	yes	characteristic
87	C₂³×C₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₃×C₄	—	2	4	no	—
89	C₂×C₃×C₄	#106	1	2	yes	—
90	C₂×C₃×C₄	#106	1	2	yes	—
91	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₃×C₄	#124	1	1	yes	characteristic
93	C₂×C₃×C₄	#124	1	1	yes	characteristic
94	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
95	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
97	C₂×Dic₃	—	8	4	no	—
98	C₂×Dic₃	—	8	2	no	—
99	C₃×C₈	—	8	2	no	—
100	C₃×C₈	#107	1	2	yes	—
101	C₃×C₈	#107	1	2	yes	—
102	C₃×C₈	#117	1	2	yes	—
103	C₃×C₈	#125	1	2	yes	—
104	C₂²×C₄	—	12	2	no	—
105	C₂²×C₄	#215	1	2	yes	—
106	C₂²×C₄	C₂×#67	1	1	yes	characteristic
107	M₁₆	—	4	2	no	—
108	M₁₆	—	4	2	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
111	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
112	C₂×C₈	#213	1	2	yes	—
113	C₂×C₈	#216	1	2	yes	—
114	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
115	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
116	C₂×C₈	#237	1	2	yes	—
117	C₂×C₈	#239	1	2	yes	—
118	C₄⋊C₄	—	12	2	no	—
119	C₄⋊C₄	—	12	2	no	—
120	C₄⋊C₄	—	24	4	no	—
121	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
122	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
123	C₄²	#214	1	2	yes	—
124	C₁₆	—	6	4	no	—
125	C₂²×C₃	—	2	2	no	—
126	C₂²×C₃	C₂×#106	1	1	yes	characteristic
127	C₂²×C₃	#486	1	1	yes	characteristic
128	C₂²×C₃	#487	1	1	yes	characteristic
129	C₃×C₄	—	2	2	no	—
130	C₃×C₄	#365	1	4	yes	—
131	C₃×C₄	#382	1	1	yes	characteristic
132	C₃×C₄	C₂×#107	1	1	yes	characteristic
133	C₃×C₄	#486	1	1	yes	characteristic
134	C₃×C₄	#487	1	1	yes	characteristic
135	Dic₃	—	16	2	no	—
136	C₂³	C₂×#215	1	1	yes	characteristic
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	24	4	no	—
139	C₂×C₄	—	24	2	no	—
140	C₂×C₄	#808	1	2	yes	—
141	C₂×C₄	#808	1	2	yes	—
142	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
143	C₂×C₄	C₂×#213	1	1	yes	characteristic
144	C₂×C₄	C₂×#214	1	1	yes	characteristic
145	C₂×C₄	#936	1	1	yes	characteristic
146	C₂×C₄	C₂×#216	1	1	yes	characteristic
147	C₂×C₄	#947	1	1	yes	characteristic
148	C₈	—	8	2	no	—
149	C₈	#807	1	2	yes	—
150	C₈	#807	1	2	yes	—
151	C₈	#890	1	2	yes	—
152	C₈	#892	1	2	yes	—
153	C₂×C₃	—	2	1	no	—
154	C₂×C₃	—	2	1	no	—
155	C₂×C₃	#1834	1	1	yes	characteristic
156	C₂×C₃	C₂×#365	1	2	yes	—
157	C₂²	—	2	2	no	—
158	C₂²	C₂×#808	1	1	yes	characteristic
159	C₂²	#5079	1	1	yes	characteristic
160	C₂²	#5088	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	48	2	no	—
163	C₄	#4494	1	4	yes	—
164	C₄	C₂×#807	1	1	yes	characteristic
165	C₄	#4977	1	1	yes	characteristic
166	C₄	#5070	1	1	yes	characteristic
167	C₄	#5093	1	1	yes	characteristic
168	C₃	#12886	1	1	yes	characteristic, complemented, Sylow
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#69574	1	1	yes	characteristic
172	C₂	C₂×#4494	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#808
commutator	C₂×C₃×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₃×#1848	C₂
socle	C₂²×C₃	C₂×#106
prime radical	C₂³×C₃	C₂×SD₁₆
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	#1848	D₃
hypocenter	C₃	#12886
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
14. #4461	C₄	semidirect
15. C₄×C₃⋊C₁₆	C₄	semidirect
168. C₃	#12886	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#1848	2
#69764	2
C₄×#812	2
#67730	2
#12886	3